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On the concavity of t-norms and triangular functions. (Spanish. English summary) Zbl 0567.26010

In this paper the author proves the following Theorem: The unique 1/2- concave function \(T: [0,1]\times [0,1]\to [0,1]\) increasing in each variable which satisfy \(T(1,x)=T(x,1)=x\) for every \(x\in [0,1]\) is \(M(x,y)=Min(x,y).\) Let us consider the set \(\Delta^+=\{F| F: {\bar {\mathbb{R}}}\to I,\quad F(0)=0,\) F is increasing and continuous to the left\(\}\). For a fixed number \(a\in {\mathbb{R}}_+\) the function \(\epsilon_ a\) is defined by \(\epsilon_ a(x)=0\) if \(x\leq a\) and \(\epsilon_ a(x)=1\) if \(x>a\). Let be \(\tau: \Delta^+\times \Delta^+\to \Delta^+\) a continuous function, increasing in each variable which satisfy \(\tau (F,\epsilon_ o)=\tau (\epsilon_ 0,F)=F\) for every \(F\in \Delta^+\) and \(\tau (\epsilon_ a,\epsilon_ a)=\epsilon_ a\) for every \(a\in [0,1]\). Using the above mentioned result the author shows that \(\tau\) is concave if and only if \(\tau =\pi_ M\) where \(\pi_ M(F,G)(x)=Min(F(x),G(x)).\)
Reviewer: D.Andrica

MSC:

26B25 Convexity of real functions of several variables, generalizations
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